COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT


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1 COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, crossclssified risk dt re compred. They re the method of BileySimon, the method of mrginl totls nd mximum likelihood method. The methods re pplied to number of risk dt sets nd compred with respect to blnce nd goodnessoffit. Multiplictive models, Triff structures. KEYWORDS The setting is s follows. l. INTRODUCTION () For certin insurnce portfolio we hve t our disposl number of rting fctors or triff rguments U, V, W,... E.g. in motor portfolio we could hve U = the ge ofthe cr, V = the home district of the owner of the cr, W = the milege of the cr per policy yer. (b) Ech tritt rgument ssumes finite number of vlues or levels, which my be denoted by consecutive integers so tht U=,2,...,mu V=l,2,..., W=l,2,...,mw The insurnce portfolio is thus divided into mo etc. m mumom w" disjoint clsses or cells, cell being defined s ll members of the portfolio corresponding to certin combintion of levels of the triff rguments. E.g., with 4 ge clsses, 7 home districts nd 5 milege clsses, we will in the exmple mentioned bove hve mu=4, too=7, row=5 nd m=490. A typicl cell will be denoted by c. The corresponding vlues of U, V, W,... will lwys be denoted by i,j, k,... respectively. (c) For ech cell c we hve observed risk dt consisting of n exposure nc (e.g., number of policy yers or insurnce sum under risk)  reltive risk mesure Pc consisting of clims totl for the cell (e.g., number of clims or clims mount) divided by the exposure ASTIN BULLETIN Vol 6, No I
2 64 AJNE (d) To the observed reltive risk mesures p~ we wnt to fit numbers fc of the form L = pu,v~wk..., ff = ~ n~pc/~ n~. In other words, we wnt to impose multiplictive triff structure on the insurnce portfoho. Putting ll u,, vj, Wk... equl to one yields the simplest possible triff structure, where ll cells get the sme premmm,5. 2. METHODS COMPARED Formlly, the multiplict~ve structure hs m,+mo+mw+. prmeters u,, vj, Wk,... i.e., one for ech possible level of the triff rguments. The number of free prmeters must, however, be less. This is shown by the fct tht we cn multiply ll uvlues nd divide ll vvlues by the sme positive number without ffecting the set bf numbers fc. The number of free prmeters is l +(m,  l)+ (too  ) + (m,. ) +. i.e., one prmeter for the overll level nd the remining ones for the "reltivitms" u2/ul, U3/Ul., V2/Vl, V3/Vl,..., W2/WI, W3/WI... This is considerbly less thn m, m the exmple given bove 24 s compred to m = 490, so perfect fit of the fc to the Pc cnnot be expected. This, on the other hnd, is of course the very ide in introducing triff structure. It should grdute, i.e., simplify, observed risk dt. In the following we will study three methods of fitting the multiplictive structure to the observed risk dt. The first two of these hve lso been discussed by VAN EEGHEN, NIJSSEN nd RUYGT (982). Reference my lso be mde to VAN EeGHEN, GREUP nd NIJSSEN (983) nd to the further references given there. We will now briefly describe the three methods for estimting the prmeters 4~, Oj, Wk,... () Mimmum ChzSqure (or BdeySimon): Put chisqure =~c nc(pcfc):/f~ The prmeters re determined so tht chlsqure is minimized. Mrginl Totls The prmeters re determined so tht [ njc=e,,cpc M M for ech mrginl M, i.e., for ech fixed i, ech fixed j, ech fixed k,... In the cse of numbers of clims s observed clims totls in the cells, nd under the ssumption tht they re stochsticlly independent nd Poissondistributed with respective prmeters nffc, this coincides with the method of mximumlikehhood (JuNG, 968).
3 TARIFF STRUCTURE AND RISK DATA 65 In hndy, menuoperted APLprogrm t our disposl we hve included third method, MLspecil This is the mximumlikelihood method under the ssumption tht the reltive risk mesures re independently distributed ccording to the norml distribution nd tht E(pc)=fc, Vr(pc)=o'2fdn~. Here g2 is n unknown proportionlity fctor common to ll cells. It my be noted tht in the cse of Poisson distributed numbers of clims, referred to bove, these equtions hold with cr2=l. They lso imply tht chisqure/m is n unbised estimtor of cr 2, chisqure being defined under () bove. In the cse of observed clims mounts, generted s sums of independent nd identiclly distributed individul clims, the equtions bove will hold true i.e., if the number of individul clims is Poisson distributed nd if the sizes of the individul clims re independent of the number of clims nd if Vr (Xc) = E(Xc)(o "2 E(Xc)) where Xc denotes typicl individul clims size for cell c. if the number of individul clims is deterministic nd if Vr (X~) = o'2e(xc). By using method we should mximize the likelihood function corresponding to the ssumptions mde. This is equivlent to minimizing chisqure/o'2+ m log g2+~c logfc with respect to u,, vj, Wk... nd o'. 3. RESULTS The tble in the ppendix summrizes some experience in using the bove mentioned methods to fit multiplictive triff structures to observed risk dt. The first column gives brief description of the risk dt, the number of triff rguments, the number of levels of ech rgument nd the totl number of cells. In column "Ct" n "" denotes tht clims mounts re observed nd "n" denotes tht clims numbers re observed (only two cses). The "Size req" sttes how mny of the observed mrginl totls do not fulfil the size requirement ncpc 9 chisqure/m. M The totl number of mrginl totls is given within prenthesis. The size requirement is very pproximte rule of thumb. It expresses the desire tht ech observed mrginl totl be equl to t lest three times its estimted stndrd
4 66 ~JNE devition ccording to the model under bove. In the size requirement the chisqure of method is thus used. For mterils not nlyzed by method, the chisqure of method ws used insted. The three colums heded "Blnce" re computed from quotients S = E nffc/~ ncpc where the estimted prmeter vlues re inserted into ft. Ech Svlue thus is quotient between grduted nd obseryed clims totls. The Svlues re computed for ll mrginls (the lrgest nd the smllest mrginl S is given in the tble) nd for the whole mteril ("Totl"). The remining four columns describe the goodnessoffit. Vr red (vrince reduction) is computed, for mrginls nd the totl s ~nc(p~f~)2/ n~(pc/~) 2 i.e., gives the vrince reduction reltive to the structure with ll cell premiums equl to p. The column "chi2" refers to vlues of chisqure, computed by inserting estimted prmeter vlues into f~. The vlues for methods nd re compred to the vlue for method (), which is the minimum vlue under the multiplictive structure. If clims numbers re observed nd if they re independent nd Poisson distributed with prmeters ccording to the multiplictive structure, minimum chisqure is for lrge exposures pproximtely chisqure distributed with degrees of freedom equl to the number of cells minus the number of free prmeters. This my be used to investigte deprtures from the hypothesis of multiplictive structure. It cn be proved tht for method () Minimum chisqure = 2(Totl S ) x ~ n~p~ so tht investigtions bsed on minimum chisqure my s well be bsed on the totl blnce of this method. 4. DISCUSSION The generl impression is tht method is the best one of those three studied. It is, by its very definition, mrginlly nd totlly blnced. It gives vrince reductions superior to method () nd superior or equivlent to those of method. Of course, it gives higher chisqure thn does method (), but where the difference is gret, the ltter method tends to show disturbing lck of blnce. Method sometimes hs smller chisqure thn method. It is lso blnced for the totl, but it my give, occsionlly, rther low blnce vlues for mrginls with smll clims totl. Method () lwys hs mrginl nd totl blnce greter thn or equl to one. The sfety mrgin tends to be lrger for mrginls with smll clims totls. An ppeling feture of this method is the erlier mentioned possibility to interpret its (lck of) totl blnce s mesure of the deprture from the multiplictive structure.
5 APPENDIX Mteril m Ct Size Req Method Mrginl Mx Min Blnce Vr Red % Totl Lest Mrg Totl Chl2 Relt. to ). Motor, glss dmge 4x7x5=490 2 Motor, hull 3xTx5= Motor, hull 7x5xT=245 4 " 5 Motor, hulllrge vehicles 2x2x3=2 6. Property, contents 5x4x6= 20 7 Property, buildings 5x4x6= Property, buddlngsfire 7x2x3x 0x6x8= Property, bridingsnot fire 7x 2x3x lox6x8=20 60 l0 Motor, motorcycles 6x 0=60 fl! n 0(26) 0(25) 0(9) 0(9) 3(6) i(5) 2(5) 0(36) 0(36) (6) () () () () () () () () () (I) ! I I 0 97 I i ! "' "t Z "...I
6 68 AJNE As support of the propositions bove it my be mentioned tht for mterils 4 nd 0 mx mrginl blnce for method () nd min mrginl blnce for method occur t mrginis with clims totls rnging from one to six percent of the verge clims totl for mrginls belonging to the sme triff rgument. Except for mteril No. 2, the mrginl concerned is the sme for both methods. Also the low vlue for method in mteril No. 6 occurs for very smll risk group. Method lso seems to be less sensitive to outlying observtions. Actully, one of the motor mterils hd such n observtion (cused by input error). After correction it turned out tht this hd disturbed the results of methods () nd much more thn tht of method. As to the mterils, the motor dt 4 re wellbhved nd show good fit to the multiplictive structure. The three methods lso generlly gve very similr results for the reltive sizes of the fctor prmeters. Mteril No. 0 is tken from FOLKESSON, NEUHAUS nd NORBERG (985). It lso shows n cceptble fit to the multiplictive structure. For mterils No. 3 nd 0 the totl number of clims is nd 3027 respectively. Minimum chisqure is 284 (d.f.=228) nd 5 (d.f.=45). The vlues re significnt on the respective levels % nd 0.05% giving rther strong evidence ginst n exct multiplictive Poisson model. The remining mterils do not behve tht well. This my be explined by () the size requirement is not fulfilled, nd/or (b) more dngerous distributions of individul clims sizes, nd, in cses 7 nd 8, (c) the very drstic reduction in the number of free prmeters, from cells to = 3 free prmeters in the multiplictive structure. This results in very low vrince reduction reltive to the single premium structure, which still my be significnt s judged by n Ftest. 6. ACKNOWLEDGEMENT Thnks re due to K. Andersson, P. Crlsson, E. Elvers nd H. Wide, who rn the progrm on risk dt of different ges nd types, nd to H. Westin, who helped to compile the ppendix. REFERENCES VAN EEGHEN, J, GREUP E K nd NIJSSEN, J A (983) Rte Mking, Surveys of Acturil Studies No 2 NtionleNederlnde N.V Rotterdm. VAN EEGHEN, J, NUSSEN, J A. nd RUYGT, F. A. M. (982) Interdependence of Risk Fctors Apphctlon on some Models In New Motor Rting Structure m the Netherlnds ASTINgroup Nederlnd FOLKESON, M., NEUHAUS, W. nd NORERG, R. (985) A Hierrchicl Credibility Model Apphed to Insurnce dt Pper presented t the 8th ASTIN Colloqmum t Blmtz JUNGG, J. (968) On Automobde Insurnce Rte Mking, ASTIN Bulletin 5 (I) B. AJNE Skndi, Sveviigen 44, S Stockholm, Sweden.
Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
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